Regular function

In algebraic geometry , a regular function is a function of some variety in its body . The ring of regular functions can be defined on any open set of the variety. These rings form a sheaf . The ring of functions that are regular throughout the variety is called the coordinate ring . Regular functions are used, among other things, to define morphisms of varieties. Regular functions are not to be confused with regular mappings, which is sometimes used in the literature to denote morphisms of varieties.

In addition, the term regular function is also used in function theory , where it denotes holomorphic functions that are not singular.

Regular functions

If it is a quasi-affine variety , then a function is regular at a point if there is an open neighborhood with and polynomials such that nowhere has zeros and . ${\displaystyle Y\subset \mathbb {A} _{k}^{n}}$ ${\displaystyle f\colon Y\to k}$ ${\displaystyle P\in Y}$ ${\displaystyle U}$ ${\displaystyle P\in U}$ ${\displaystyle g,h\in k[x_{1},\ldots ,x_{n}]}$ ${\displaystyle h}$ ${\displaystyle U}$ ${\displaystyle f|_{U}={\tfrac {g}{h}}}$

Is a quasi-projective variety, then a function is regular at a point if there is an open neighborhood (in the Zariski topology ) with and there are homogeneous polynomials with the same degree, so that nowhere has and . ${\displaystyle Y\subset \mathbb {P} _{k}^{n}}$ ${\displaystyle f\colon Y\to k}$ ${\displaystyle P\in Y}$ ${\displaystyle U}$ ${\displaystyle P\in U}$ ${\displaystyle g,h\in k[x_{0},\ldots ,x_{n}]}$ ${\displaystyle h}$ ${\displaystyle U}$ ${\displaystyle f|_{U}={\tfrac {g}{h}}}$

${\displaystyle g}$and are not functions on that , but is a well-defined function since and are homogeneous of the same degree. ${\displaystyle h}$ ${\displaystyle \mathbb {P} _{k}^{n}}$ ${\displaystyle {\tfrac {g}{h}}}$ ${\displaystyle g}$ ${\displaystyle h}$

If a quasi-affine or a quasi-projective variety, then a function is regular if it is regular at every point in . ${\displaystyle Y}$ ${\displaystyle f\colon Y\to k}$ ${\displaystyle Y}$

If the body is identified with the affine space , then a regular function is continuous in the Zariski topology. ${\displaystyle k}$ ${\displaystyle \mathbb {A} _{k}^{1}}$

An important consequence of this results for irreducible varieties: If and are regular functions on and if there is a non-empty open set on which and match, then and on match. Because the set of all points on which is is not empty, closed and tight. ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle Y}$ ${\displaystyle U\subset Y}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle Y}$ ${\displaystyle f-g=0}$

The sheaf of regular functions and the coordinate ring

For every open set , the set of regular functions forms a ring, which is denoted by. These rings form a groove . Since the regular functions are defined by local properties, they even form a sheaf . This sheaf is closely related to the affine scheme of the variety. The ring of functions that are regular across the variety is called the coordinate ring . He is isomorphic to . It is the Verschwindeideal of , so the ideal of polynomials at each point of zero. ${\displaystyle U\subset Y\subset \mathbb {A} _{k}^{n}}$ ${\displaystyle U}$ ${\displaystyle {\mathcal {O}}(U)}$ ${\displaystyle A(Y)}$ ${\displaystyle k[X_{1},\ldots ,X_{n}]/I(Y)}$ ${\displaystyle I(Y)}$ ${\displaystyle Y}$ ${\displaystyle Y}$

The coordinate ring is a domain of integrity and a finitely generated algebra. ${\displaystyle A(Y)}$ ${\displaystyle k}$

The local ring of a point

The local ring of a point is the ring of seeds of regular functions. This ring is referred to as or only . This ring therefore consists of equivalence classes of with , where is equivalent to if and on match. This ring is a local ring , its maximum ideal consists of the germs that disappear in. ${\displaystyle {\mathcal {O}}_{P,Y}}$ ${\displaystyle {\mathcal {O}}_{P}}$ ${\displaystyle \langle U,f\rangle }$ ${\displaystyle P\in U}$ ${\displaystyle \langle U,f\rangle }$ ${\displaystyle \langle V,g\rangle }$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle U\cap V}$ ${\displaystyle P}$

literature

• Klaus Hulek : Elementary Algebraic Geometry . Vieweg, Braunschweig / Wiesbaden 2000, ISBN 3-528-03156-5 .
• Robin Hartshorne : Algebraic Geometry , Springer-Verlag, New York / Berlin / Heidelberg 1977, ISBN 3-540-90244-9
• Ernst Kunz : Introduction to Commutative Algebra and Algebraic Geometry , Vieweg (1980), ISBN 3-528-07246-6